Question: Let $w,$ $x,$ $y,$ and $z$ be nonnegative numbers whose sum is 100.  Find the largest possible value of
\[wx + xy + yz.\]
We have that
\[wx + xy + yz \le wx + xy + yz + zw = (w + y)(x + z).\]By AM-GM,
\[(w + y)(x + z) \le \left( \frac{(w + y) + (x + z)}{2} \right)^2 = 2500.\]Equality occurs when $w = x = 50$ and $y = z = 0,$ so the largest possible value is $\boxed{2500}.$